Kolmogorov-type inequalities for mixed derivatives of functions of many variables
Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{r\gamma}_{\infty}(T^d...
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| Datum: | 2004 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2004
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| author | Babenko, V. F. Korneichuk, N. P. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. |
| author_facet | Babenko, V. F. Korneichuk, N. P. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:08:43Z |
| description | Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{r\gamma}_{\infty}(T^d)D^{r\gamma}x\neq0}
\frac{||D^{k\gamma}x||_{L_{\infty}(T^d)}}{||x||^{1-k/r}||D^{r\gamma}||^{k/r}_{L_{\infty}(T^d)}} = \infty$$
and $$||D^{k\gamma}x||_{L_{\infty}(T^d)} \leq K||x||^{1 - k/r}_{L_{\infty}(T^d)}||D^{r\gamma}x||_{L_{\infty}(T^d)}^{k/r}
\left(1 + \ln^{+}\frac{||D^{r\gamma}x||_{L_{\infty}(T^d)}}{||x||_{L_{\infty}
(T^d)}}\right)^{\beta}$$
for all $x \in L^{r\gamma}_{\infty}(T^d)$
Moreover, if \(\bar \beta \) is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$ |
| first_indexed | 2026-03-24T02:48:37Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-3779 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:48:37Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cc/291c89b97204b7e50a472e71ee1d91cc.pdf |
| spelling | umjimathkievua-article-37792020-03-18T20:08:43Z Kolmogorov-type inequalities for mixed derivatives of functions of many variables Неравенства типа Колмогорова для смешанных производных функций многих переменных Babenko, V. F. Korneichuk, N. P. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{r\gamma}_{\infty}(T^d)D^{r\gamma}x\neq0} \frac{||D^{k\gamma}x||_{L_{\infty}(T^d)}}{||x||^{1-k/r}||D^{r\gamma}||^{k/r}_{L_{\infty}(T^d)}} = \infty$$ and $$||D^{k\gamma}x||_{L_{\infty}(T^d)} \leq K||x||^{1 - k/r}_{L_{\infty}(T^d)}||D^{r\gamma}x||_{L_{\infty}(T^d)}^{k/r} \left(1 + \ln^{+}\frac{||D^{r\gamma}x||_{L_{\infty}(T^d)}}{||x||_{L_{\infty} (T^d)}}\right)^{\beta}$$ for all $x \in L^{r\gamma}_{\infty}(T^d)$ Moreover, if \(\bar \beta \) is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$ Нехай $γ = (γ_1 ,..., γ_d )$ — вектор з додатними координатами, $D^γ$— відповідна мішана похідна (порядку $γ_j$- з а $j$-ю змінною). Доведено, що при $d > 1$ і довільних $0 < k < r$ $$\sup_{x \in L^{r\gamma}_{\infty}(T^d)D^{r\gamma}x\neq0} \frac{||D^{k\gamma}x||_{L_{\infty}(T^d)}}{||x||^{1-k/r}||D^{r\gamma}||^{k/r}_{L_{\infty}(T^d)}} = \infty$$ Разом з тим для всіх $x \in L^{r\gamma}_{\infty}(T^d)$ $$||D^{k\gamma}x||_{L_{\infty}(T^d)} \leq K||x||^{1 - k/r}_{L_{\infty}(T^d)}||D^{r\gamma}x||_{L_{\infty}(T^d)}^{k/r} \left(1 + \ln^{+}\frac{||D^{r\gamma}x||_{L_{\infty}(T^d)}}{||x||_{L_{\infty} (T^d)}}\right)^{\beta}$$ Більш того, якщо \(\bar \beta \) —найменше можливе значення показника Р в цій нерівності, то $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$ . Institute of Mathematics, NAS of Ukraine 2004-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3779 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 5 (2004); 579-594 Український математичний журнал; Том 56 № 5 (2004); 579-594 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3779/4279 https://umj.imath.kiev.ua/index.php/umj/article/view/3779/4280 Copyright (c) 2004 Babenko V. F.; Korneichuk N. P.; Pichugov S. A. |
| spellingShingle | Babenko, V. F. Korneichuk, N. P. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. Пичугов, С. А. Kolmogorov-type inequalities for mixed derivatives of functions of many variables |
| title | Kolmogorov-type inequalities for mixed derivatives of functions of many variables |
| title_alt | Неравенства типа Колмогорова
для смешанных производных функций многих переменных |
| title_full | Kolmogorov-type inequalities for mixed derivatives of functions of many variables |
| title_fullStr | Kolmogorov-type inequalities for mixed derivatives of functions of many variables |
| title_full_unstemmed | Kolmogorov-type inequalities for mixed derivatives of functions of many variables |
| title_short | Kolmogorov-type inequalities for mixed derivatives of functions of many variables |
| title_sort | kolmogorov-type inequalities for mixed derivatives of functions of many variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3779 |
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